Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
Using the integral
Z 1
0
e 2 cos d ˆ 1 r
e 2 =4 ;
2
we ®nd
T…x; t† ˆ 1 Z 1
Z 1
p
2 f …u†e …ux†2 =4kt du f …u†e …u‡x†2 =4kt du :
kt 0
0
p
Letting …u x†=2
p
kt ˆ w in the ®rst integral and …u ‡ x†=2 kt ˆ w in the second
integral, we obtain
"
T…x; t† ˆp
1 Z 1
p
p
f …2w Z 1
kt ‡ x†dw ew2 p
f …2w
#
p
kt x†dw :
ew2
kt
x=2
x=2
kt
Fourier transforms for functions of several variables
We can extend the development of Fourier transforms to a function of several
variables, such as f …x; y; z†. If we ®rst decompose the function into a Fourier
integral with respect to x, we obtain
f …x; y; z† ˆp
1
2
Z 1
1
…! x ; y; z†e i! xx d! x ;
where is the Fourier transform. Similarly, we can decompose the function with
respect to y and z to obtain
with
f …x; y; z† ˆ 1 Z 1
g…!
…2† 2=3 x ;! y ;! z †e i…! xx‡! y y‡! z z† d! x d! y d! z ;
1
g…! x ;! y ;! z †ˆ 1 Z 1
f …x; y; z†e i…! xx‡! y y‡! z z† dxdydz:
…2† 2=3 1
We can regard ! x ;! y ;! z as the components of a vector ! whose magnitude is
q
! ˆ ! 2 x ‡ ! 2 y ‡ ! 2 z;
then we express the above results in terms of the vector !:
f …r† ˆ 1 Z 1
g…x†e ixr dx;
…2† 2=3 1
g…x† ˆ 1 Z 1
f …r†e …ixr† dr:
…2† 2=3 1
…4:44†
…4:45†
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